Calculations

Calculations

• Skyline Cholesky factorization

  Factorizes the stiffness of the whole model once for every load set.

  K = L Lᵀ; L_ii = √(K_ii − Σ L_ik²)

  Frame analysis with slabs, walls and rigid floors frame/plate-solve.js

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• Forward and back substitution

  Solves the factorized system for one load vector.

  L y = f; Lᵀ x = y

  Frame analysis with slabs, walls and rigid floors frame/plate-solve.js

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• Loads on plates

  Turns a plate's own weight and area loads into forces at its corners.

  corner force = (∫ Nᵢ dA) × p; own weight p = −factor × f × ρ × 1e-9 × g × t

  Frame analysis with slabs, walls and rigid floors frame/plate-solve.js

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• Node displacements from the solution

  Gives every node its six displacements, including those tied to a floor's master point.

  u_q = Σ c × x_e over the constraint map of q

  Frame analysis with slabs, walls and rigid floors frame/plate-solve.js

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• Plate forces into the reactions

  Adds each plate's corner forces to the support reactions and stores the plate's center results.

  f = K_g u, added at the corners; center results at ξ = η = 0

  Frame analysis with slabs, walls and rigid floors frame/plate-solve.js

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• Plate results at the nodes

  The moments and forces shown at each node of a slab or wall.

  inside: mean of the corner values of the plates at the node; on an edge: a + b x + c y fitted through nearby Gauss values, read at the node

  Frame analysis with slabs, walls and rigid floors frame/plate-solve.js

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• Least-squares plane through Gauss values

  The value at a node on a slab's edge, from a plane fitted through the integration point values around it.

  normal equations 3 × 3, first unknown by Cramer's rule

  Frame analysis with slabs, walls and rigid floors frame/plate-solve.js

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• Story drift

  How far each rigid floor moves relative to the floor below, at its center and at its worst node.

  drift = move(floor) − move(floor below) at the same point; ratio = |drift| ÷ story height

  Frame analysis with slabs, walls and rigid floors frame/plate-solve.js

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• Combined plate results

  Adds plate results and floor movements of several load sets, each times its factor.

  result = Σ fᵢ × resultᵢ

  Frame analysis with slabs, walls and rigid floors frame/plate-solve.js

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• Cleaning an outline

  Removes repeated points and corners that lie on a straight run, and makes the outline counter-clockwise.

  drop a point when |p − q| ≤ 1e-6; drop a corner when |cross| ≤ 1e-9 × |b − a|²

  Meshing a slab or wall into plates frame/plate-mesh.js

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• Is the outline square-cornered

  Finds the turn that makes every edge of the outline and its openings run along one of two square directions.

  turn = atan2 of the longest edge; each edge: min(|dx|, |dy|) ≤ 1e-6 × max(1, length)

  Meshing a slab or wall into plates frame/plate-mesh.js

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• Grid mesh

  Lays grid lines through every corner and given point and divides each gap into equal parts no longer than the size asked for.

  n = ⌈gap ÷ size − 1e-9⌉; line_j = a + (b − a) × j ÷ n

  Meshing a slab or wall into plates frame/plate-mesh.js

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• Point inside a polygon

  Decides whether a grid cell's center lies inside the outline or inside an opening.

  even–odd crossing count

  Meshing a slab or wall into plates frame/plate-mesh.js

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• Mesh of a four-corner outline

  A structured mesh between the opposite edges of a convex four-corner outline.

  P(s, t) = (1 − s)(1 − t) P₀ + s (1 − t) P₁ + s t P₂ + (1 − s) t P₃; s = i ÷ n, t = j ÷ m

  Meshing a slab or wall into plates frame/plate-mesh.js

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• Cutting an outline into triangles

  Ear clipping of an outline that is neither square-cornered nor four-cornered.

  an ear: cross(a, b, c) > 1e-12 and no other corner inside

  Meshing a slab or wall into plates frame/plate-mesh.js

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• Triangles into quadrilaterals

  Cuts each triangle into three quadrilaterals and each of those into k × k plates.

  k = ⌈longest edge ÷ 2 ÷ size − 1e-9⌉; nodes by bilinear blend

  Meshing a slab or wall into plates frame/plate-mesh.js

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• Mesh quality

  The largest side ratio and the smallest corner angle of the plates.

  aspect = longest side ÷ shortest side; angle = acos(u·v ÷ (|u| |v|))

  Meshing a slab or wall into plates frame/plate-mesh.js

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• Mesh nodes onto model nodes

  Uses an existing node of the frame for a mesh node that lies within the merge tolerance of it, else adds a new node.

  |existing − new| ≤ tol (1 mm by default)

  Meshing a slab or wall into plates frame/plate-mesh.js

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• Wall plates

  Meshes a vertical wall along a line in plan between two heights.

  node = from + e_x × u, z = z₀ + v; e_x = (to − from) ÷ L

  Meshing a slab or wall into plates frame/plate-mesh.js

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• Is a plate a proper quadrilateral

  Refuses a plate whose corners are in a line or folded over.

  each corner: (e₁ × e₂) · n ÷ |n| > 1e-3 × |e₁| × |e₂|

  Meshing a slab or wall into plates frame/plate-model.js

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• Nodes of a rigid floor by height

  Collects the nodes that belong to a floor given by its height.

  |z_node − z_floor| ≤ tol (1 mm by default)

  Meshing a slab or wall into plates frame/plate-model.js

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• Bar positions of a beam

  Places the top, bottom and side bars of a beam section from their count and cover.

  y_i = −b ÷ 2 + c + (b − 2c) × i ÷ (n − 1); side bars evenly between the rows

  Concrete section by strain compatibility frame/rc-section.js

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• Bar positions of a column cage

  Places the bars around the perimeter of a column, corners counted once.

  y_i = −y_c + 2 y_c × i ÷ (n − 1); y_c = b ÷ 2 − cover

  Concrete section by strain compatibility frame/rc-section.js

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• Bars gathered into layers

  The depth of each bar from the compression face, with bars at one depth added into one layer.

  d = depth ÷ 2 − sign × position, rounded to 1e-6 mm

  Concrete section by strain compatibility frame/rc-section.js

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• Axial force and moment for a neutral axis depth

  The section's axial force and its moment about mid-depth when the neutral axis is at depth c.

  a = min(β₁ c, h); C_c = α₁ f′c b a; ε_s = ε_cu (c − d) ÷ c; f_s = clamp(E_s ε_s, ±f_y); P = C_c + Σ F; M = Σ F × (h ÷ 2 − d)

  Concrete section by strain compatibility frame/rc-section.js

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• Neutral axis for a given axial force

  The neutral axis depth at which the section's axial force equals the target.

  bisection on c between 1e-6 × depth and 1000 × depth

  Concrete section by strain compatibility frame/rc-section.js

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• Stress block factor β₁

  The depth of the equivalent stress block as a share of the neutral axis depth.

  0.85 for f′c ≤ 28; 0.85 − 0.05 (f′c − 28) ÷ 7; 0.65 for f′c ≥ 55

  Concrete member strength to ACI 318-19 frame/rc-aci318.js

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• Strength reduction factor

  The factor φ for axial force and bending from the strain in the extreme tension steel.

  0.90 for ε_t ≥ ε_ty + 0.003; 0.65 (0.75 spiral) for ε_t ≤ ε_ty; a straight line between

  Concrete member strength to ACI 318-19 frame/rc-aci318.js

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• Square root of the concrete strength in shear

  The √f′c used in the shear formulas.

  min(√f′c, 8.3)

  Concrete member strength to ACI 318-19 frame/rc-aci318.js

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• Bending strength of a beam section

  The nominal and design moment of a section at zero axial force, with the minimum steel and ductility tests.

  c from P(c) = 0; Mn = M(c); φMn; As,min = max(0.25 √f′c ÷ fy, 1.4 ÷ fy) b d

  Concrete member strength to ACI 318-19 frame/rc-aci318.js

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• Steel required for a moment

  The steel area a singly reinforced section needs for a factored moment.

  Rn = Mu ÷ (φ b d²); As = (0.85 f′c ÷ fy) (1 − √(1 − 2 Rn ÷ 0.85 f′c)) b d

  Concrete member strength to ACI 318-19 frame/rc-aci318.js

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• One-way shear strength

  The concrete and stirrup contributions to shear strength, the minimum stirrups and the spacing limit.

  Vc by Table 22.5.5.1 (a), (b) or (c); λs = min(1, √(2 ÷ (1 + 0.004 d))); Vs = Av fyt d ÷ s; φVn = 0.75 (Vc + Vs)

  Concrete member strength to ACI 318-19 frame/rc-aci318.js

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• Axial force and moment diagram

  The nominal and design strength outline of a column section for bending about one axis.

  (Mn, Pn) at 121 depths c from 20 h down to 0.001 h, plus the balanced, tension-controlled and P = 0 points; Po = 0.85 f′c (Ag − Ast) + fy Ast

  Concrete member strength to ACI 318-19 frame/rc-aci318.js

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• Utilization on the diagram

  How far a factored axial force and moment are along the ray from the origin to the diagram.

  ratio = √(Mu² + Pu²) ÷ t, t = where the ray meets the outline

  Concrete member strength to ACI 318-19 frame/rc-aci318.js

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• Threshold torsion

  The torque above which torsion would have to be designed for.

  φ Tth = 0.75 × 0.083 λ √f′c Acp² ÷ pcp

  Concrete member strength to ACI 318-19 frame/rc-aci318.js

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• Strengths of a section, worked out once

  The bending strengths for both axes and signs, the two strength diagrams, the effective shear depths and the threshold torsion of a section.

  shear depth d = centroid of the bars beyond mid-depth, or 0.8 × depth when there are none

  Concrete member and slab checks on a solved frame frame/rc-design.js

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• Design moment strength at an axial load

  The largest design moment the strength diagram allows at a given axial force.

  linear interpolation between the two diagram points that bracket Pu

  Concrete member and slab checks on a solved frame frame/rc-design.js

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• Moment magnification of a slender column

  The magnified design moment of a non-sway column about one axis.

  Pc = π² (0.4 Ec Ig ÷ (1 + βdns)) ÷ (k L)²; δ = Cm ÷ (1 − Pu ÷ 0.75 Pc) ≥ 1; Mc = δ max(M₂, Pu (15 + 0.03 h)); Ec = 4700 √f′c

  Concrete member and slab checks on a solved frame frame/rc-design.js

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• Drawn section against the concrete section

  Checks that the analyzed section's area and inertias agree with the b × h the bars were given for.

  |A − b h| ≤ 0.05 b h; I within 5 % of b h³ ÷ 12 × 1, 0.70 or 0.35

  Concrete member and slab checks on a solved frame frame/rc-design.js

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• Beam flexure ratio

  The utilization of a beam in bending about one or both axes.

  |My| ÷ φMny + |Mz| ÷ φMnz

  Concrete member and slab checks on a solved frame frame/rc-design.js

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• Column axial force and bending ratio

  The utilization of a column under axial force with one or two moments.

  one moment: the ray ratio on its diagram; two, Pu ≥ 0.1 f′c Ag: max(r_y + r_z − Pu ÷ (φ Po), Pu ÷ φPn,max); two, lower Pu: |My| ÷ φMny(Pu) + |Mz| ÷ φMnz(Pu)

  Concrete member and slab checks on a solved frame frame/rc-design.js

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• Design moments of the slab's bar layers

  The moment each of the four bar layers must carry so that bending and twisting are both covered.

  M*x = Mx + |Mxy|, M*y = My + |Mxy|; if one is negative: 0, and the other raised by Mxy² ÷ |that moment|

  Concrete member and slab checks on a solved frame frame/slab-design.js

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• Slab reinforcement and shear

  The steel required and provided per meter in each layer, and the one-way shear check of a slab.

  d = h − cover − bar ÷ 2 (outer layer); provided = (π d_b² ÷ 4) × 1000 ÷ s; need = max(required, ρ_min × 1000 × h); ratio = need ÷ provided

  Concrete member and slab checks on a solved frame frame/slab-design.js

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• Concrete stiffness for the analysis

  The elastic modulus, shear modulus and density given to concrete members.

  E = 4700 √f′c; G = E ÷ 2.4; ρ = 2400 kg/m³

  Concrete member and slab checks on a solved frame frame/frame-concrete.js

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• Cracked stiffness factor

  The share of the gross stiffness used for cracked members.

  0.70 columns; 0.35 beams; 1 when not cracked

  Concrete member and slab checks on a solved frame frame/frame-concrete.js

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• Area of a bar

  The cross-section area of a reinforcing bar from its diameter.

  A = π d² ÷ 4

  Concrete member and slab checks on a solved frame frame/frame-concrete.js

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• Approximate period

  The building's approximate fundamental period from its height and structural system.

  Ta = Ct × (hn ÷ 1000)^x, hn in mm

  Seismic loads and story drift to ASCE 7-22 frame/loads-asce7.js

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• Upper limit factor on the period

  The factor Cu that caps a period taken from analysis.

  table of SD1 against Cu, straight lines between rows

  Seismic loads and story drift to ASCE 7-22 frame/loads-asce7.js

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• Seismic response coefficient

  The coefficient Cs that multiplies the building's weight to give the base shear.

  Cs = SDS ÷ (R ÷ Ie), at most SD1 ÷ (T R ÷ Ie) or SD1 TL ÷ (T² R ÷ Ie), at least max(0.044 SDS Ie, 0.01) and 0.5 S1 ÷ (R ÷ Ie) where S1 ≥ 0.6

  Seismic loads and story drift to ASCE 7-22 frame/loads-asce7.js

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• Height exponent

  The exponent k of the vertical distribution of the story forces.

  k = 1 for T ≤ 0.5 s; 2 for T ≥ 2.5 s; 1 + (T − 0.5) ÷ 2 between

  Seismic loads and story drift to ASCE 7-22 frame/loads-asce7.js

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